There are conventionally known automatic transmissions in which an automatic speed change occurs based on the travelling speed of the vehicle. In any of these transmissions, however, a first speed stage is established when the vehicle starts to move. With an increase in vehicle speed, an up-shift operation occurs in sequence to a second and a third speed stage.
During the starting of the vehicle in motion on a low .mu. road (.mu. is a friction coefficient) such as a snowy road, a frozen road or a muddy road, if the acceleration pedal is depressed in order to start the vehicle in the same manner as at the starting on a normal road, the driving power becomes too large when compared with the frictional force between the tires and the road surface, so that the tires may slip in some cases.
In this case, the vehicle may be started in motion smoothly by slightly depressing the acceleration pedal with the brake pedal being remained depressed at the starting of the vehicle and gradually reducing the force for depressing the brake pedal in this state to gradually transmit the driving power to the tire. In conventional automatic transmissions, however, the first speed stage is established when the vehicle is started so that driving power varies largely depending upon the degree of depression of the acceleration pedal. Therefore, delicate operation of the brake pedal and the acceleration pedal are required in order to smoothly start the vehicle.